Stochastic single-machine scheduling with random resource arrival times

  • Lianmin Zhang
  • Yizhong Lin
  • Yujie Xiao
  • Xiaopeng Zhang
Original Article
  • 143 Downloads

Abstract

Scheduling in an uncertain environment remains a meaningful yet challenging direction of research. In this paper, we consider a new scheduling setting from practical complex business applications, where resources (e.g., raw materials) used for processing jobs arrive randomly, due to reasons such as unstable transportation service caused by extreme weather conditions, unreliable suppliers, unpredictable industrial actions, etc. Further, jobs must be processed one by one and preemption is not allowed. The processing times of jobs are not known but their distribution. We incorporate these factors into a stochastic single-machine scheduling model and examine two different common types of objectives: minimizing total expected weighted completion time and minimizing total expected weighted squared completion time. We derive and prove a natural and intuitive optimal policy for the model with the first objective. Besides, we find that, under some mild conditions, the well-known policy in stochastic scheduling, WSEPT (weighted shortest expected processing time), still holds optimal for achieving either of objectives. The numerical example further supports and illustrates our results, which provide decision-makers insights into tricky uncertain scheduling problems.

Keywords

Stochastic scheduling Completion times Squared completion time 

Notes

Acknowledgements

This work was partially supported by the National Natural Science Foundation of China (Grant No. 71501093, 71501090), the Basic Research Foundation (Natural Science) of Jiangsu Province (Grant No. BK20150566).

References

  1. 1.
    Bagga PC, Kalra KR (1981) Single machine schednling problem with quadratic function of completion times-a modified approach. J Inf Optim Sci 2:103–108MATHGoogle Scholar
  2. 2.
    Birge J, Frenk JBG, Mittenthal J, Rinnooy Kan AHG (1990) Single-machine scheduling subject to stochastic breakdown. Nav Res Logist 37:661–677MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Cai XQ, Zhou X (1999) Stochastic scheduling on parallel machine subject to random breakdowns to minimize expected costs for earliness and tardy cost. Oper Res 47:422–437MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Cai XQ, Zhou X (2000) Asymmetric earliness-tardiness scheduling with exponential processing times on an unreliable machine. Ann Oper Res 98:313–331MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Cai XQ, Sun X, Zhou X (2004) Stochastic scheduling subject to machine breakdowns: the preemptive-repeat model with discounted reward and other criteria. Nav Res Logist 51:800–817Google Scholar
  6. 6.
    Cai XQ, Wu X, Zhou X (2009) Stochastic scheduling on parallel machines to minimize discounted holding costs. J Sched 12:375–388MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cai XQ, Wu XY, Zhang L, Zhou X (2014) Scheduling with stochastic approaches. In: Sotskov U, Werner F (eds) Sequencing and scheduling with inaccurate data, Nova, New YorkGoogle Scholar
  8. 8.
    Glazebrook KD (1991) On nonpreemptive policies for stochastic single-machine scheduling with breakdowns. Probab Eng Inf Sci 5:77–87MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Glazebrook KD (2005) Optimal scheduling of tasks when service is subject to disruption: the preempt-repeat case. Math Methods Oper Res 61:147–169MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Lee C-Y, Yu G (2007) Single machine scheduling under potential disruption. Oper Res Lett 35:541–548MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Li W, Braun WJ, Zhao YQ (1998) Stochastic scheduling on a repairable machine with Erlang uptime distribution. Adv Appl Probab 30:1073–1088MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Mittenthal J, Raghavachari M (1993) Stochastic single machine scheduling with quadratic early-tardy penalties. Oper Res 41:786–796MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Ning Y, Chen X, Wang Z, Li X (2016) An uncertain multi-objective programming model for machine scheduling problem. Int J Mach Learn Cyber. doi: 10.1007/s13042-016-0522-2 Google Scholar
  14. 14.
    Pinedo M, Rammouz E (1988) A note on stochastic scheduling on a single machine subject to breakdown and repair. Probab Eng Inf Sci 2:41–49CrossRefMATHGoogle Scholar
  15. 15.
    Qi XD, Yin G, Birge JR (2000a) Scheduling problems with random processing times under expected earliness/tardiness costs. Stoch Anal Appl 18:453–473MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Qi XD, Yin G, Birge JR (2000b) Single machine scheduling with random machine breakdowns and randomly compressible processing times. Stoch Anal Appl 18:635–653MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Rothkopf MH (1966) Scheduling independent tasks on parallel processors. Manag Sci 12:437–447MathSciNetCrossRefGoogle Scholar
  18. 18.
    Rothkopf MH (1966) Scheduling with random service times. Manag Sci 12:707–713MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Singh A, Valente JMS, Moreira MRA (2012) Hybrid heuristics for the single machine scheduling problem with quadratic earliness and tardiness costs. Int J Mach Learn Cybern 3(4):327–333CrossRefGoogle Scholar
  20. 20.
    Smith WE (1956) Various optimizers for single-stage production. Nav Res Logist Q 3:59–66MathSciNetCrossRefGoogle Scholar
  21. 21.
    Townsend W (1978) The single machine problem with quadratic penalty function of completion times: a branch-and-bound solution. Manag Sci 24:530–534CrossRefMATHGoogle Scholar
  22. 22.
    Zhou X, Cai XQ (1997) General stochastic single-machine scheduling with regular cost functions. Math Comput Model 26:95–108MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Lianmin Zhang
    • 1
  • Yizhong Lin
    • 2
  • Yujie Xiao
    • 3
  • Xiaopeng Zhang
    • 4
  1. 1.School of Management and EngineeringNanjing universityNanjingChina
  2. 2.School of BusinessJiaxing UniversityJiaxingChina
  3. 3.Jiangsu Key Laboratory of Modern LogisticsSchool of Marketing and Logistic Management, Nanjing University of Finance and EconomicsNanjingChina
  4. 4.School of Business AdministrationZhejiang Gongshang UniversityHangzhouChina

Personalised recommendations